12

Linear homopolar synchronous motor for urban transit application

Moteur synchrone homopolaire et lineaire pour application sur vehicules de transport urbain

By G.R. Slemon, FCSEE, University of Toronto, Toronto, Ontario and ft.P. Bhatia, BBC Brown Boveri Canada Ltd., Toronto, Ontario.

A linear homopolar synchronous motor suitable for application in tracked urban transit vehicles is described, analyzed and tested as a small machine in a rotating test wheel. In addition to providing propulsion which is independent of adhesion limitations, this motor promises to provide better efficiency and power factor than other types of linear motor and thus to require a smaller on-board inverter. Potential economics in track construction are also envisaged. The paper presents some design information and an equivalent circuit model o f the motor.

Un moteur synchrone homopolaire et lineaire, pour application sur vehicules de transport urbain sur voies, est decrit, analyse et mis a l'epreuve comme une petite machine dans une roue rotative d'essai. En plus de fournir une force de propulsion, independante des limites d'adherence, ce moteur sera caracterise par une plus grande efficacite et un facteur de puissance plus eleve que ceux d'autres types de moteur lineaire et, par conse­quent, requerra a bord un convertisseur plus petit. On envisage egalement un certain potentiel d'ordre economique dans la construction de voies. Cette etude presente certains renseignements quant a la conception et au modele de circuit equivalent du moteur.

Introduction

In some urban transit situations, propulsion by driven steel wheels on steel rails may be unsatisfactory because of the limita­tion of adhesion on steep grades or in maintaining short headway in wet or icy conditions. What is required is a means of propulsion which does not rely on adhesion. Linear electric motors provide such a means by exerting a driving force directly upon an air gap between motor and track-mounted reaction rail. Until recently, at­tention was concentrated mainly on the linear induction motor . Experience has shown that this motor , while effective in providing propulsion, is inherently inefficient and operates at a relatively low power factor, thus requiring a large on-board power conditioner.

This paper describes work carried out on an alternative propul­sion device, the linear homopolar synchronous motor . It covers briefly the design of the motor , the development of an electrical equivalent circuit, the prediction of performance, results of tests to determine the accuracy of the circuit model and results of tests on a small motor to determine the accuracy of performance predic­tion. The paper concludes with an assessment of the potential of this motor for urban transit propulsion.

Description

The linear homopolar synchronous motor (LHSM), first pro­posed by H. K e m p e r , 1 2 is distinguished by having both its ar­mature and field windings on the vehicle interacting with track-mounted steel poles as shown in Figure 1. The armature has two side-by-side sections of laminated iron joined by a laminated yoke. A three-phase winding is fitted into slots in the two sections. Field coils, fitted above the three-phase winding, produce flux down one armature section, across the air gap to the staggered poles oppos­ing that section, across the track yoke to the poles opposing the other armature section, across the second air gap and up the other armature section. The field produced by the track pole structure and the travelling wave produced by the three-phase ac currents in the armature winding interact to produce thrust . The field in the air gaps also attracts the armature toward the track.

The three-phase ac winding of the L H S M is supplied from a cur­rent source inverter whose output frequency can be varied upward from zero. A variety of means may be employed to maintain syn­chronism between the motor velocity and the inverter frequency, such as pole position detection or power factor cont ro l . 3 4

FIELD COILS

Figure 1: Arrangement of the LHSM as seen from the air gap.

Can. Elec. Eng. J. Vol 7 No 2, 1982

S L E M O N / B H A T I A : S Y N C H R O N O U S M O T O R 13

showed that no significant increase in the useful component of flux density was produced by making the pole depth D greater than the pole gap S. In most instances, it was shown to be adequate to make D > 055.

t — T — L

Figure 2: Longitudinal section of LHSM.

The maximum flux density B in the air gap above each pole is limited by saturat ion in the armature teeth. Figure 3 shows the ratio of the fundamental component of air gap flux density B , to B for a range of dimension ratios X /G ' and 77 X, a s s u m i n g / ) / 5 > 1. It will be noted tha t the op t imum ratio of pole width to wavelength is in the range 0.35-0.4 from the standpoint of maximum thrust per unit of gap area.

Analysis of air gap field

The first step in the design of the LHSM is to determine the shape of the track pole to produce opt imum thrust. Figure 2 shows a longitudinal view of the motor as seen from one side and simplified by ignoring the slotting of the armature and by replacing the pole structure of Figure 1 with poles projecting upward from a ferromagnetic plate. The choice of a rectangular pole, rather than a tapered pole as proposed by Levi et al, was based on a preliminary flux analysis which snowed that , because of flux fring­ing, the flux at the pole base was significantly greater than at the pole top surface. 5 6 7 The full width of pole was considered to be necessary to avoid saturation at the pole base.

The air gap length G is determined by consideration of mechanical clearance and is typically 10-15 mm for urban vehicles. The effective gap length G' takes into account the effect of arm­ature slotting using the conventional Carter coefficient.

Thrust is maximized when the pole width T is chosen with respect to the wavelength X to give the maximum alternating com­ponent of longitudinal air gap flux density. The analytical ap­proach employed conformal mapping using the Schwartz-Christoffel t ransformation coupled with numerical i n t e g r a t i o n . 9 1 0

An important design choice is the depth D of the pole side, since this depth should be minimized to reduce track mass. The analysis

B

0 2 M .6 -8 Figure 3: Effect of pole width T on fundamental flux density.

-T/X

B ' •6

0 2 -4 ' * 8 Figure 4: Effect of pole width T on average flux density.

10- -T/X

]Q-AXIS

D-a^cis D-apcis D - a x i s

o m r£ T3 r$ rg r? T§ T9 i7o~

T/X

Figure 5: Variation ofBld and BXq as a function of gap dimensions.

The mass of the L H S M is dominated by the mass of the iron structure, a large part of which is in t h e yoke and in the section above the a rmature windings. The cross-sectional areas of these sections are chosen to accommodate the air gap flux with an allowance for leakage. The average value B of the air gap flux den­sity is then a measure of the dominant mass o f t h e moto r . Figure 4 shows the rat io of the average flux density of B t o S for a range of pole dimension rat ios. It is noted that reduction in pole width results in reduced average flux density. Thus , depending on overall design criteria, the value of pole width may be chosen somewhat

14 C A N . E L E C . E N G . J. VOL 7 N O 2, 1982

A r m a t u r e y o k e

L e a k a g e f lux ,<fy

\ Gap f r i n g i n g flux,<fy

A i r g a p f l u x +J

Figure 6: Field winding flux components.

- @ ( V W ? ( W W - — ( g ) -

RI.

- W W V

R, < R ,

Figure 7: Magnetic circuit.

Values of the reluctances in Figure 7 may be found by conformal mapping in the case of fringing fluxes and by suitable simplifying approximations in the case of field leakage flux. The overall circuit may then be used to determine the flux in each iron member when the armature teeth are near saturation, and propor t ion these iron members accordingly.

Electrical equivalent circuit

The L H S M may be represented by the same form of electric equivalent circuit as is appropriate for a salient-pole rotat ing syn­chronous machine, as shown in Figure 8 . 1 1 In this circuit, Rs is the a rmature resistance per phase. The armature leakage inductance U arises from slot leakage and end winding leakage for which stan­dard methods of calculation may be employed. A n additional leakage component arises from the straight part of the armature winding between the two armature sections.

The direct and quadra ture axis magnetizing inductances are given by

Lmd — BxdAgapNsel Ia

and

L<mq — B\qAgapNsel Ia

(1)

(2)

where Bld and Blq are as shown in Figure 5, Agap is the total ar­mature air gap area, Nse is the equivalent number of a rmature turns per pole and Ia is the armature current per phase.

less than the opt imum shown in Figure 3. For example, with X / G ' = 40, a reduction in 77X from 0.4 to 0.3 results in a thrust reduc­tion of about eight per cent and a reduction in flux, and, therefore, a reduction in yoke mass, of about 18 per cent.

A magnetic field is also produced in the air gap by the three-phase armature currents and this field links with the armature win­ding. The upper inset in Figure 5 shows the D-axis condition when the pole centre coincides with the peak value of the armature mmf, while the lower inset shows the Q-axis condition where pole centre coincides with the zero value of armature mmf. Analysis of the field in these two conditions, again using conformal mapping, gives the alternating components of air gap flux density Bld and Blq

(as ratios of the flux density Bmax for T = X), as shown in Figure 5 for a range of dimension ratios. It is noted that Bld = Blq for values of 77X in the range 0.35-0.5. This suggests that reluctance effects (such as reluctance thrust) are likely to be negligible in a L H S M for the pole dimension ratios normally employed.

Magnetic equivalent circuit

Design of the iron structure of the motor requires a knowledge of the flux in each section. Figure 6 shows the various flux paths in a cross-section of the motor . The yoke consists of the main air gap flux <t>g plus the gap fringing flux <f>f and the field leakage </>,. The various reluctance paths offered to the three flux components are shown in the magnetic equivalent circuit of Figure 7, where F i s the mmf of one field coil, Ry is the nonlinear reluctance of half the yoke, Ri is half the reluctance of the field leakage pa th , Rg is the reluctance of one air gap, Rf is the reluctance of the fringing flux path on one side of the pole, and Rc is the reluctance of half of the track rail and pole.

The air gap reluctances on the two sides of the motor are not always equal because of the finite length of the armature . In a short machine, when the pole area under one side may be significantly different from that under the other, the flux <t>g may be modulated. However, in a practical motor , the number of poles under the armature length will usually be sufficient to make this ef­fect negligible.

The current ratio n' in Figure 8 relates the field current // to the equivalent ac current source lr'. With no load on the motor , the load angle 6 is zero. With Ia = 0, the induced emf in the motor is

Em =jo)Lmdn'If (3)

This induced emf may also be determined from the rms value of the ac component of air gap flux density Bt as shown in Figure 3 using the equation

Em=juBxAgapnse (4)

Combining Eq . 3 and 4, a value of the current ratio n' can be com­puted.

As noted previously, the direct and quadrat ic magnetizing in­ductances are nearly equal for the usual choice of pole dimension ratios. Thus , the reluctance branch in the equivalent circuit of Figure 8, involving the difference between these two values, may normally be eliminated. If, in addit ion, the armature resistance Rs

is ignored for purpose of circuit calculatuon, the simplified circuit of Figure 9 may be derived. In this circuit

Ld = Li + Ln

and

n — -— n Ed

(5)

(6)

: I A J x l

.Xmd .Xmq { J X m d - X m q

( X m d . X m q ) c t &

v X m d - X m q '

U j l r . n ' . I f Z z i

Figure 8: Equivalent circuit.

S L E M O N / B H A T I A : S Y N C H R O N O U S M O T O R 15

The circuit of Figure 9 may be used to calculate the power Ps for values of armature current /«, field current / / and load angle 6. The motor thrust may then be computed from the expression

f=Pslv

where v is the motor velocity.

(7)

L H S M test equipment

Tests were performed on a small LHSM arranged as shown in Figure 10. A 2-m diameter wheel was fitted with a solid iron rim carrying 56 solid iron poles bolted alternately on each side of the rim. An arc-type armature was suspended under the test wheel on six force transducers. These transducers along with an electronic conversion unit permit the measurement of the three forces and their moments on the armature.

F O R C E TABLE

SIDE VIEW END VIEW

Figure 10: Arrangement of the system under test.

A current-source inverter was used to supply the armature and control the magnitude, frequency and phase angle of the armature current. The control system is shown in Figure 11. Six optical sen­sors on the test wheel frame detect the pole position. The six thyristors in the inverter are then fired at a controlled delay time after the sensor signal. Control of this delay time permits precise control of the phase angle 5.

The shaft of the test wheel was belt connected to a 20 -kW dc machine which was used to drive or load the L H S M . The air gap of the motor was continuously monitored to compensate for any changes which might result from changes in the attractive force between the armature and the wheel. The dimensions of the test system are given in Table 1.

L m d / L J r L - 6

= l r " L - 5

C h o p p e r C h o p p e r

i i F r e q u e n c y a n d

P h a s e C o n t r o l

A r m a t u r c

^ P o l e

^ o s i t i o n

S e n s o r

F i r i n g P h a s e

C i r c u i t s D e l a y

Figure 11: Control system.

G=10mm

0 1 2 3 U 5 6 7 9 10 11 12 13 14 15 16 17 1ft 19 20 21 22

^TOOTH NUMBER

Figure 9: Simplified equivalent circuit. Figure 12: Average armature flux density distribution.

T A B L E 1 D i m e n s i o n s o f test m o t o r ( m m )

Armature Track Stack Width 55 Pole length T 109, 88 length 441 Interpolar span S 111.8, 132.8

Pole width 55 Number of poles 3 Wavelength X 220.8 Yoke width 110 Slots/pole-phase 2 Yoke thickness 30 Slot width 9.2 Number of poles 56 Sloth depth 38 Wheel diameter 1968 Coil pitch 92 Turns /coil 8 Air gap Resistance/phase 0.555G Field turns/coil 453 Length G 5, 10, 15 Mass 93kg.

\ \

\ \

(V-|S31)A

lISN3C

]dV9 3A

V

I n d u c t o r

—nn^-

16 CAN. E L E C . E N G . J. VOL 7 N O 2, 1982

— I f ( A M P )

Figure 13: Maximum and minimum values of armature flux density as a function of field current.

Test results

Figure 12 shows the modulat ion of gap density for several values of field current in the test motor . The flux density varies linearly with field current except at high values of // where the ar­mature teeth reach saturation. As a result of this saturat ion, the maximum density 6 saturates while the minimum density B con­tinues to rise proportionately to // as shown in Figure 13. It is noted that the difference 6- B reaches a maximum for a particular value of //, showing that this value of // produces maximum modulat ion.

Figure 14 shows the value of flux at various parts of the motor as a function of //. The leakage fluxes are seen to be large in com­parison with rotating synchronous machines because of the relatively large air gap. Figure 15 shows the attractive force as a function of // for three values of gap length. The curves are of square law form at low values of // but become more nearly linear due to saturation. The air gap field caused by armature current is shown in Figure 16 for the direct axis positioning of the poles and in Figure 17 for the quadratic axis positioning.

/ ^ . Y O K E

As a pole enters the front end of the a rmature , the establishment of flux is opposed by eddy currents in the solid iron of the pole. Similarly, the decay of flux is inhibited by eddy current as the pole leaves the back end of the a rmature . The test moto r was made in­tentionally short (i.e. three pole) to accentuate this end effect. Figure 18 shows the induced voltage in search coils in the armature teeth along the armature length for various values of speed. It is noted tha t , as speed encreases, the search coil emf continues to in­crease further back in the motor . The effect is not large, however, even at the speed of 24m / s, a reasonable maximum value for urban vehicles, suggesting that solid iron poles may be satisfactory for full scale motors with large numbers of poles.

The eddy current losses in the motor with no a rmature current are shown in Figure 19 for various values of field current and speed. While the losses are significant for the small test machine, it should be noted that they arise from an end effect which will re­main constant as the length of the motor is increased for practical application. If necessary, a substantial reduction in eddy current losses could be achieved by splitting each pole into two sections.

Figures 20 and 21 show the open circuit a rmature voltage per unit of frequency and the armature short circuit current respective­ly for a range of values of field current and speed. These curves show both the effect of saturation and the effect of eddy current in

Figure 15: Vertical force versus field current.

Figure 14: Motor flux as a function of field current. Figure 16: Variation of armature flux density (in D~axis).

S L E M O N / B H A T I A : S Y N C H R O N O U S M O T O R 17

IF=A0Amp

Figure 17: Variation of armature flux density (in Q-axis).

LONGITUDINAL DISTRIBUTION OF INDUCED VOLTAGE

-4 -Direction of motioi

If=30A

G=10mm

V ( n r y s ) Figure 19: Eddy current losses as a function of speed and field current.

5 1 0 Coil number 1 5

Figure 18: Longitudinal distribution of induced voltage.

the magnetic system. Using the results of Figures 20 and 21 at low speed and in the unsaturated region, the parameters of the equivalent circuit model of Figure 9 were evaluated and compared with values calculated from field analysis. The results are shown in Table 2.

Figure 22 shows the motor thrust as a function of current angle 8 for various values of field current, both as measured and as calculated from the equivalent circuit using no load test da ta and using calculated parameters from field anaysis. It is noted tha t good agreement is achieved at low values of // but tha t the accuracy of prediction becomes much poorer as saturation occurs. These curves show the expected result that maximum thrust per unit of armature current occurs at a current angle of 90° . The sinusoidal shape of the curves demonstrates that there is negligible saliency or reluctance thrust . It is also noted that the driving and regenerating characteristics are essentially identical. With the current angle set at 90°, Figure 23 shows the effect of speed on maximum thrust for various values of speed, gap length and armature current. The ef­fect of saturation is noted.

With the current angle set for maximum thrust , the L H S M was tested for a wide range of load conditions in both motor ing and generating regions. Typical results of input power, mechanical power efficiency and power factor are shown in Figure 24, together with results predicted for the equivalent circuit using no bad test da ta and using analytically determined parameters . It should be noted that the analytical model neglects both saturat ion and eddy current effects. It is seen that the predicted results are in reasonable agreement with test results. However, bo th the predicted efficiency and the predicted power factor are higher than obtained experimentally. The discrepancies are at tr ibutable to ig­noring the effects of eddy currents and space harmonics and also to simplifying assumptions made in the air gap analysis.

Figure 20: Open circuit flux linkage as a function of field current.

T A B L E 2 Equiva lent circuit parameters

Test Analysis Synchronous inductance (mH) 1.60 1.69 Current ratio 1.58 1.75

POLE P 0 L E

18 C A N . E L E C . E N G . J. VOL 7 N O 2, 1982

Figure 21: Short circuit armature current versus field current.

G = 10mm I, = 15 amp. v = 10 m/s

predicted from no-load results

Test results.

The test results of Figures 12 to 24 were obtained with a pole length of 109 m m As a test of the air gap analysis, the pole length was reduced to 88 m m . retaining the same wavelength. Experimen­tal results generally confirmed the analysis and were otherwise similar to those with the longer poles.

Design considerations

The test motor used in this study was designed to provide ex­perimental information for the development of an analytical model rather than t o demonstrate the capabilities of this type of moto r . The small number of poles was chosen to accentuate the end effects including the eddy current losses. As a small machine of only a few kilowatts capacity, its efficiency would be expected to be significantly lower than a moto r for transit application of 200-30 k W capacity. Figure 24 shows that the efficiency can be high at low values of a rmature current . In a large motor , the relative value of armature resistance is reduced and , thus , high ef­ficiency can be expected over the whole range of applicable ar­mature currents . The eddy current losses are due to an end effect and remain substantially constant as the length of the machine is increased from three to typically 15-30 poles in a full-sized motor . The effect of these losses on large moto r efficiency may then be ex­pected to be relatively small even with the continued use of simple solid iron poles.

While the maximum speed in transit applications is likely t o be in the range 20-25m / s, maximum thrust need be produced only up to a break point speed usually of the order 10m/s . It will be noted from Figure 19 that the eddy current losses are relatively low at this break point speed. At higher values of speed, the eddy current losses can be minimized by reducing the field excitation.

The power factor of the test motor is observed to be high at low values of a rmature current but to decrease rapidly as current is in­creased. In the small machine, the end winding leakage is a major contr ibut ion to the inductance of the motor . In a large motor , the width of each armature section would be increased but no increase in the dimensions of the end windings is necessary. Thus , reduc­t ion in the relative magni tude of leakage inductance and increase in power factor is expected in a full scale moto r . Figure 25 shows a graph of the predicted power factor of the moto r as a function of the ratio of a rmature to effective field current for various values of the ratio of leakage inductance to magnetizing inductance. It is seen that reduction of the leakage inductance to 80 per cent of the magnetizing inductance can result in a power factor over 60 per cent at full load and even to a leading power factor at low load.

(a)

— • . . . . ^ .

20.0

0 i

... / . ^ V - - - - - -

- : \ |--r-S . ..; . . . ;

ac

3= L i

\[// ;• • •'• *\\\ ;

G = 10mm la = 50 amp v = 10 m/s

60.00 110.00 220.00 [DECREES)

' p r e d i c t e d f rom analyt ical values of parameters.

500

400

T h r u s t (Nv\|

300

200

100

I f = 1 5 a m p . /

/ ' 1 0 m / s | / / G=5mm

20 m/s I

^ 1 0 m/s 1 G = 10mm

^ 2 0 m/s J

10 m/s 1 G = 1 5 m m

20 m/s J

20 40 60 80 100 > — I a(amp)

Figure 22: Thrust as a function of field angle». Figure 23: Effect of speed on thrust.

S L E M O N / B H A T I A : S Y N C H R O N O U S M O T O R 19

LEADING

G = 10mm l/ = 15 amp. v = 10 m/s

predicted f rom no-load results.

Test results.

(a)

G = 10mm la = 50 a m p V = 10 m/s

predicted from analyt ical values of parameters.

Test results.

(b)

Figure 24: Machine performance characteristics as a function of armature current.

P.F.

LAGGING

Figure 25: Effect of leakage inductance on power-factor.

count provides an accurate measure of vehicle position. It is an­ticipated, therefore, that systems with L H S M drive can be design­ed to provide highly reliable short headway operat ion.

The choice of an all steel track provides promise of avoiding track distortion due to temperature change and thus permit minimum design air gap. For urban speeds up to 100km/h, it not anticipated that any part of the track structure need be laminated. It is expected that the track can be factory assembled with ade­quate accuracy of pole positioning in 10-15m lengths. The choice of a track system with staggered poles on opposite sides of the track yoke allows use of simple rectangular ac coils in the armature structure. A n alternative track structure makes use of side-by-side poles but requires more complex figure-of-eight armature coils which are also more difficult to cool.

The product of eff ic iency and power factor is a measure of the relation between the mechanical power rating of the motor and the volt-ampere rating of its on-board power converter. Because of space limitations on transit vehicles, minimization of the power converter rating has become a critical design consideration. It is noted from Figure 24 that a power factor-efficiency product in ex­cess of 0.35 is achieved in the small test motor up to about 60 per cent of its maximum test load. For a full scale motor , this product is expected to be significantly higher over the whole load range. In comparison, full scale linear induction motors operated in transit vehicles with the same air gap rarely exceed a power efficiency pro­duct of 0.35.

Conclusion

The linear homopolar synchronous motor shares with the linear induction motor the features of no contact propulsion, no moving parts and freedom from the grade acceleration and braking restric­tions imposed by wheel adhesion. The results of this study suggest that the L H S M may have an advantage over the LIM in requiring a significantly smaller volt-ampere capacity in its accompanying on­board inverter.

An additional potential advantage of the L H S M arises in the control of vehicle position and headway. Since the vehicle moves one wavelength for each cycle of the ac supply, the supply frequen­cy provides an accurate measure of vehicle speed and the cycle

The equivalent circuit model presented in this paper for the linear homopolar synchronous motor has been shown to give a reasonably accurate prediction of the performance of a small test motor . While improvements in the analysis are desirable, it is ex­pected that the present model will provide results adequate for the design of a full scale machine. The analysis also provides useful in­formation on the choice of the dimensions and proport ions of the track and armature structures.

A potential disadvantage of the L H S M is the strong attractive force between the armature structure and the track. This force adds to the wheel loading and may cause flexure of the track beam. The small test motor produced a thrust density up to 10kN/m 2 of air gap surface using current values which were well within convec­t ion cooling limits. A fullscale urban transit motor capable of 20kN thrust at 10m/s might be expected to have an air gap surface somewhat less than 2 m 2 in view of the increased current loadings normally associated with larger machines. This study suggests that the L H S M may be a strong competitor for urban transit propul­sion in the future and justifies further development and testing.

Acknowledgement

The work reported in this paper was supported by a contract from the Transporta t ion Development Centre of Transpor t Canada , Contract N o . 050780-00311, and was reported in Transpor t Canada Report N o . TP2856E (Jan . 1981) entitled " S t u d y of an Analytical Model ;of the Homopola r Linear Syn­chronous M o t o r " .

20 C A N . E L E C . E N G . J. VOL 7 N O 2, 1982

References

1. H . Kemper , "Elektr i sch A n g e Triebene E i senbahn- fahrzeuge mit Elektromagnetischer Schwebefuhrung," ETZ pp. 11-14, 1953.

2. E. Rummich, 4 'Machines Liniaires Synchrones, theorie et realization pratiques," Bull ASE, Vol. 63, pp. 1338-44, Nov. 1972.

3 . Canadian Maglev Group, "Superconducting Linear Synchronous Motor Propul­sion and Magnetic Levitation for High Speed Ground Transportation," Cana­dian Institute for Guided Ground Transport, Kingston, Ontario, Phase II, Report N o . 75-5 , March, 1976.

4. G.R. Slemon, " A Homopolar Linear Synchronous Motor ," Electr. Machines Electromech. Vol. 4, pp. 59-70, July-August, 1979.

5. E. Levi, " A Preliminary Evaluation of Electrical Propulsion by Means of Iron Cored Synchronously Operating Linear Motors ," report on DOT-FRR, NTIS N o . PB-2586437, Jan. 1975.

6. E . Levi, L. Birenbau and Z. Zabar, "Concerning the Design of Inductor Syn­chronous Motors Fed by Current Source Inverter," IEEE Trans, on Magnetics, Vol. M A G - 1 3 , N o . 5, pp. 1421-1423, Sept. 1977.

7. E . Levi, J .P . Lee, F. Lalezari and M.S . Gemelos, "Computer Aided Conformal Mapping of Magnetic Fluxes in Saturated Inductor Motors ," IEEE Trans. Magnetics, Vol. M A G - 1 4 , N o . 5, September, 1978, Intermag Conference, Florence, Italy, pp. 927-928, May 9-12 , 1978.

8. N . Boldea and S.A. Naser, "Field Winding Drag and Normal Forces on Linear Synchronous Homopolar Motors ," Electr. Mach. Electromech, Vol. 2, N o . 3, pp. 253-268, April-June 1978.

9. V. Murthy, "Improvements in Numerical Techniques for Conformal Transfor­mation with Particular Reference to Electro-magnetic Fields ," IEEE Trans, on Magnetics, M A G - 1 4 , pp. 233-240, July 1978.

10. R .P . Bhatia, "Modelling of a Linear Homopolar Synchronous Machine", M.A.Sc . Thesis, Department of Electrical Engineering, University of Toronto, October, 1980.

11. G.R. Slemon, Magnetoelectric Devices, John Wiley and Sons Inc., p. 397, 1966.

University of Toronto Department of Electrical Engineering

The Department of Electr ical Engineer ing, University of Toronto, has a number of academic posi t ions available commenc ing in the 1982-1983 academic year as fo l lows:

• Assistant/Associate Professor (tenure stream) in the f ie ld of e lect ronic devices, w i th research interests in LSI/VLSI and computer-a ided des ign. Duties wi l l in­c lude undergraduate and graduate teach ing in e lect ron ics . A PhD and a substan­t ial research record are required.

• Three assistant professors in the f ie lds of: ( a ) control engineering, w i th research interests in mul t ivar iable contro l theory

and its appl icat ion to industr ia l problems. A good knowledge of micro­processors and interface e lect ron ics wou ld be advantageous.

( b ) communications, w i th research interests in data commun ica t ions or d ig i ta l signal processing. A good knowledge of d ig i ta l systems wou ld be advan­tageous.

( c ) computer engineering, w i th research interests in mul t iprocessor systems, hardware design or related areas, and the abi l i ty to teach microprocessors and computer organizat ion at the undergraduate level.

These three posi t ions are for in i t ia l terms of up to three years and are renewable. Duties include undergraduate teach ing in courses w i th intensive hardware laboratories and graduate teach ing. A PhD in e lectr ical engineer ing is required.

• One year term or visiting appointments at the assistant professor level are available in the f ie lds of ( a ) electronics—computer-aided design and modern signal processing. ( b ) computer engineering—any area.

Salaries offered wi l l be commensurate w i th qua l i f i ca t ions and exper ience. App l ican ts should send a curr icu lum vitae, w i th the names of three referees and an ind icat ion of the posi t ion being sought to:

Professor H.W. Smith, Chairman Department of Electrical Engineering University of Toronto Toronto, Ontario M5S 1A4

Appl icants wi l l be accepted unti l May 31,1982. In accordance w i th Canadian immigrat ion requirements th is advert isement is d i rected to Canadian c i t izens and permanent residents.